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\thesistitle{Etude d’un produit d’assurance paramétrique contre le risque de pluie torrentielle en Jamaïque}
\title{Pricing of a New Insurance Parametric Product Covering Losses from Extreme Rainfall}
\author{Cuong NGUYEN QUOC and Florent RITLENG}
\date{May 2014}

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\textbf{Keywords}: Extreme Rainfall, CCRIF, Extreme Value Theory, Generalized Pareto Distribution, Markov Chain, Copula, Pure Premium, Parametric Index. \\

%Les catastrophes naturelles représentent un risque particulièrement important à la fois pour les habitants et les économies de l’ensemble des pays des Caraïbes. Les pertes engendrées par ces événements rares et intenses affectent l’économie de ces pays et peuvent atteindre des montants considérables (8\% du PIB de la Jamaïque pour les dégâts causés par l’ouragan Ivan en 2004). En outre, les pays touchés n’ont parfois pas les ressources suffisantes pour faire face à ce type d’intempérie. De fait, l’enjeu le plus important consiste à trouver les besoins en liquidité à court terme nécessaires au maintien en état des infrastructures et des services publics, en attendant que des ressources supplémentaires soient disponibles pour financer des besoins de reconstruction et d’adaptation à plus long terme. C’est dans ce contexte que 16 Etats des Caraïbes, en association avec la Banque Mondiale, ont pris l’initiative en 2007 de créer un système commun d’assurance à but non lucratif. Ce système, le \textbf{Caribbean Catastrophe Risk Insurance Facility} (CCRIF), est le premier pool d’assurance multi-pays créé dans le monde, et c’est aussi le premier mécanisme d’assurance qui adosse un mécanisme de couverture paramétrique. Par rapport à une assurance classique, les indemnités ne sont pas reversées selon les dégâts réels constatés au sol mais en fonction d’un indice paramétrique facilement mesurable lors d’un événement extrême (ex : vitesse du vent, pic de précipitation). \\ 

%\noindent Le mécanisme initial du CCRIF ne couvre cependant pas un type d'événement particulièrement fréquent dans les Caraïbes : les pluies torrentielles. Ceci a conduit le CCRIF à fournir une couverture pour le risque d’inondation. Depuis Juin 2013, le CCRIF propose un contrat d’assurance appelé \textbf{CCRIF/Swiss Re Excess Rainfall} (XSR) qui offre une couverture des dommages dus à des pluies torrentielles en Jamaïque. Ce produit repose sur l’estimation d’un \textbf{indice paramétrique} reflétant l’extrême intensité des pluies torrentielles sur une courte période dans plusieurs zones géographiques du pays. Il est donc essentiel de comprendre comment est construit un tel indice afin d’identifier les facteurs de risques influençant la prime du produit. La problématique principale de nos travaux est de modéliser la dynamique de cet indice dans le but d’établir un profil de risque du pays souscripteur. L’enjeu est de pouvoir donner un prix au contrat XSR.

Natural disasters are a particularly high risk for both people and economies of the Caribbean. The losses caused by these rare and intense events affect the economy of these countries and can reach considerable amounts. (e.g. 8 percent of GDP in Jamaica for the damage caused by Hurricane Ivan in 2004). In addition, affected countries may not have sufficient resources to cope with these extreme events. In this context, 16 Caribbean States in association with the World Bank launched in 2007 the first non-profit multi-country pool called the \textbf{Caribbean Catastrophe Risk Insurance Facility }(CCRIF). Moreover, it is the first time that such a pool offers a parametric insurance coverage. Unlike conventional insurance, payments are not made according to the damage observed on the ground but according to the measure of a parametric index defined in the contract such as wind speed, rainfall amount. \\ 

However, the initial CCRIF mechanism doesn’t provide coverage of extreme rainfall that often affects the Caribbean. Thus, the CCRIF forges a partnership in June 2013 with Swiss Re to offer a new coverage for excess rainfall called \textbf{CCRIF/Swiss Re XSR}. This product relies on the estimation of a \textbf{parametric index} reflecting the intensity of extreme rainfall over a short period of time in several geographical areas of the country. Therefore, it is essential to understand how such an index is constructed to identify factors influencing the product’s premium. The main issue of our work is to model the index’s dynamics in order to establish a risk profile of the country. The challenge is to put a price on the XSR policy.


\section*{Methodology}

We first proceed by making an inventory of methodologies used in the implementation of the XSR product. We use public information provided by the CCRIF to build a parametric index aggregating rainfall of several grids of Jamaica. In other words, we define a parametric national index as the sum of local indexes. Additional assumptions are necessary to calculate an index on the 15 years-historical rainfall data. \\

Once the steps of construction of the index are defined, we attempt to model its evolution. We have adopted a progressive approach starting with a simple model of daily rainfall to a more complex model of extreme rainfall. Several statistical tools were used including Markov Chain, extreme value theory and copula. \\

After modeling the rainfall, we calculate an annual premium for the XSR product through the Monte-Carlo method. We illustrate the risk of geographic dependence of extreme rainfall by comparing local independent premiums with national premium. Finally, we analyze the product by calculating sensitivities and introduce the concepts of risk transfer.

\section*{XSR Product}

The Swiss Re/CCRIF XSR model is based on data provided by the Tropical Rainfall Measuring Mission (TRMM). This mission was designed by NASA and the Japanese Aerospace Exploration Agency to monitor and study tropical rainfall. TRMM provides a real-time satellite-based estimate of aggregate rainfall for every grids of Jamaica. A risk exposure rate is given for all the 28 grids as shown in figure \ref{fig:Repartition 28 cellules}. 
\begin{figure}[htbp]
     \centering
       \includegraphics[scale=0.45,keepaspectratio=true]{Figures/Chapter1/Cellules_TRMM_Jamaica.png}
       \rule{35em}{0.5pt}
     %\caption{Répartition des 28 cellules géographiques sur la Jamaïque}
     \caption{Jamaica divided in 28 grids}
     \label{fig:Repartition 28 cellules}
\end{figure}
\\

In the XSR contract, an extreme event and its associated parametric index are defined according to the following 4 steps: \\
\begin{enumerate}[1) ]
\item For each TRMM grid nod, a 5-day aggregate rainfall is calculated every day using a moving window. A local event occurs when the \textbf{5-day aggregate} exceeds 250 mm and ends on the day before rainfall next falls below 250 mm\footnote{5-day aggregated rainfall are defined as the sum of daily precipitation 5 days prior to the day of measurement}. %. L’événement se termine lorsque le cumul des pluies sur 5 jours redescend en-dessous de ce seuil (cf. figure  \ref{fig:XSR step 2}). \\
\begin{figure}[htbp]
   \centering
     \includegraphics[scale=0.6,keepaspectratio=true]{Figures/Chapter1/XSR_step2.png}
     %\rule{35em}{0.5pt}
   \caption{How to identify local extreme event}
   \label{fig:XSR step 2}
\end{figure}
\item For each event at each TRMM grid node, the single highest 5-day aggregate rainfall measurement is used to calculate the index loss rate via a vulnerability curve which maps loss percentage to rainfall amounts. \\
\item The vulnerability rate\footnote{ $\textit{vulnerability }(peak)=C \left(\mathbb{1}_{250<peak<B} \times \frac{peak-250}{B-250}+\mathbb{1}_{peak>B}\right)$  where B and C are adjustable parameters.} for each grid event is applied to the exposure value of the TRMM grid node, so as to construct the individual index loss for the event within the grid node. \\
\item To calculate the national index loss, the individual index losses at each grid node are added together each day. National-level events are defined as continuous periods when there is an ongoing event at one or more grids. \\
\end{enumerate}
The national index is defined as follows : 
\begin{align*}
I_{\textit{national}} = \sum_{i}^{28} I_{\textit{grid i}} = \sum_{i}^{28} w_i \times \textit{vulnerability } ( \textnormal{peak}_\textit{grid i})
\end{align*}
For each event, the amount given to the country is calculated via the following payout function :
\begin{equation*}
\textit{Payout} =
\left\{
\begin{array}{ll}
0  & \textit{if } I< \textit{Attachment}\\\\
\frac{I-\textit{Attachment}}{\textit{Exhaustion}-\textit{Attachment}}          \times \textit{Coverage Limit}  & \textit{if }\textit{Attachment} <I< \textit{Exhaustion} \\ \\
\textit{Coverage Limit} & \textit{if } \textit{Exhaustion}<I
\end{array}
\right.
\end{equation*}

Two hundred local events and 29 national events were identified over the past fifteen years.

%\section*{Modélisation statistique de l'indice paramétrique}
\section*{Statistical modeling of parametric index}
In order to set a price on XSR contract, we carefully model rainfall to predict the likely payouts and thus the contract premium.
\paragraph{Modeling of daily rainfall} The 'lowest' level of modeling the local index is based on the modeling of daily rainfall. We focus on two approaches : the Bartlett-Lewis process and modeling by Markov Chain. Both models combine an occurence process of rainy days with a pattern of rain intensity.\\

In the Bartlett- Lewis process, the starting days of rainy periods follow a Poisson process and their duration follows an exponential law. In our observation data, the time between two rainy events does not follow an exponential law, while durations of rainy event, counted in days, are difficult to be modelized by a continuous law. \\

To overcome this limitation, we model the occurrence process of raining day by a 2-state Markov chain, wet and dry states. After testing several laws for the intensities , we used the Weibull distribution to calibrate daily precipitation (see Figure \ref{fig: Weibull-Kingston}). By this method, the number of extreme events is three times less than the number of historical events. Furthermore, the peak level of extreme events is lower than the historical one. As a result, this model underestimates the extreme events which encourages us to model directly exceedances over threshold.
\begin{figure}[htbp]
\centering
%\caption[Pluie journalière en Jullet au nord Kingston]{Pluie journalière en Jullet au nord Kingston}
    \includegraphics[width=6.5cm]{Figures/Chapter2/pluie_jullet_nord_kingston.pdf}
  %\caption{Calibration des pluies journalières par la loi de Weibull}
  \caption{Calibration of daily rainfall by the Weibull}
  \label{fig: Weibull-Kingston}
\end{figure} 


\paragraph{Modeling of peak over threshold} The XSR contract is triggered only when the 5-day aggregated rainfall exceeds a certain threshold. We use the extreme value theory to calibrate the peak of exceedances independently over 28 grids. Theoretically, the aim is to estimate the parameters of $X|X>250$ where $ X $ is the 5-day aggregate of rainfall measurements. In practice, we select a threshold $u<250$ low enough to have sufficiently data. The threshold of 100mm is selected for Kingston area and applied to other grids. Given that 5-day rainfall serie is not independent, estimation data are chosen sufficiently spaced in time by the  "clustering" method. We model only the peak of exceedances on each cluster. \\ 
\begin{figure}[htbp]
\centering
  \includegraphics[scale=0.3,keepaspectratio=true]{Figures/Chapter2/clusters.pdf}
  \caption{Representation of clusters/local events for a threshold of 100mm}
\end{figure} 

According to the extreme value theory, the peaks are modeled by Generalized Pareto Distribution with two parameters ($ \sigma , \xi $). On the grid including Kingston, the average exceedance of an extreme event is 366 mm in a heavy tail model ($\xi>0$) and 328 mm in a light tail model ($\xi= 0$).\\

After estimating local indexes distribution, we model their dependence via the copula theory.

\paragraph{Multivariate modeling of extreme} Sklar's theorem allows us to obtain the multivariate distribution of $(Y_1,\dots,Y_{28} )$ from the marginal distributions of $Y_i$ and a copula function $C$ characterizing the dependence among $Y_i$. Denoting $y_i$ peak over a certain threshold of the grid $i$, one has:
\begin{equation*}
H(y_1,y_2,\dots,y_n)=C(F_1(y_1),F_2(y_2),\dots,F_n(y_n))
\end{equation*}
where \begin{itemize}
\item H is the distribution function of $(Y_1,\dots,Y_{28})$
\item $F_i$ is ditribution function of $Y_i$. \\
\end{itemize}

\begin{table}[htbp] 
\centering
\footnotesize
%\scriptsize
\begin{tabular}{l|c|c|c}
\textbf{Natural Catastrophe}&\textbf{Year}&\textbf{National Index}& \textbf{Quantile}\\
\hline \hline 
Hurricane Michelle & 2001 & 0,43 & 36,23\%\\
May/June Extreme Rainfall & 2002   & 10,96 & 86,79\%\\
Hurricane Ivan & 2004  &  4,39 & 73,16\% \\
Hurricane Wilma & 2005  & 10,17 & 85,85\%\\
Extreme Rainfall before Hurricane Dean & 2007  & 1,62 & 56,91\% \\
Tropical Storm Gustav & 2008 &   8,26 & 82,88\% \\
Tropical Storm Nicole& 2010 & 27,98 & 96,24\%\\
\hline
\hline
\end{tabular}
\caption{Distribution of extreme events in the simulated national index}
\label{tab1}
\end{table}

The dependence of rainfall over 28 areas is modeled by the normal copula for operational reasons. We simulate 10,000 times the multivariate distribution of 5-day rainfall exceedances in order to obtain the national index distribution.Table \ref{tab1} shows the distribution of historical natural disasters among simulated indexes.


\section*{Pricing of XSR product}
The annual pure premium is defined as the expectation of payouts annually paid to the subscriber. We assume that the number of annual events $N$ is independent of payouts paid to Jamaica. Therefore the annual premium is written :
\begin{equation*}
\begin{split}
\textit{National Premium}&=\mathbb{E}\left(\sum_{n=1}^{N}\textit{Payout}_n \right) \\
			  &=\mathbb{E}\left(N\right)*
			  	\mathbb{E}\left[Payout_{nat}(I_{nat})\right] 
\end{split}
\end{equation*}

As for the payout function, we set the limits Attachment and Exhaustion such as 29 historical events are covered. By simulating 10,000 times the national index, the national pure premium for a coverage limit of \$28 million over one year is \$8.68 million.\\ 

To illustrate the lack of extreme dependence in exceedance modeling, we calculate local annual premiums of local contracts which the payouts paid in each grid depend only on the local indexes. Coverage is proportional to the exposure of each cell and the total coverage amount is \$28 million. The cost of 28 local coverages is \$5.12 million.

\section*{Risk and sensitivities of XSR}
%\paragraph{Choix du coefficient de queue dans la modélisation des dépassements}
\paragraph{Choice of tail parameter in peak over threshold modeling}
%Dans le calcul de la prime pure, nous avons utilisé les modèles de dépassement à queue épaisse $\xi>0$ pour la majorité des cellules. En calibrant un modèle à queue plus légère ($\xi=0$), la prime nationale est à 6,20 millions de dollars, soit une diminution de 28\%. 
In calculating the pure premium, we used a heavy tail POT model $ \xi> $ 0 for the majority of grids. Meanwhile, with a light tail model ($\xi = 0 $), the national premium is \$6.20 million which represents a decrease by 28 \%.

%\paragraph{Choix de copule} Nous illustrons les limites de la copule gaussienne \textit{via} la comparaison du comportement des extrêmes des observations et celui du modèle gaussien. La copule de Gumbel est plus adpatée pour modéliser les extrêmes en dimension 2 mais ne représente pas une solution opérationnelle en dimension 28.
\paragraph{Choice of copula} We illustrate the limits of the normal copula by comparing the behavior of extreme values from observation data and that from the Gaussian model. Gumbel copula is more fitted to model extremes dependence in dimension 2 but does not appear as an operational solution in dimension 28.

%\paragraph{Sensibilité par rapport au seuil de sélection} Nous modifions le seuil $u$ qui est de 100 mm pour la ville de Kingston. La prime décroit avec $u$ dans un modèle à queue épaisse. 

\paragraph{Sensitivity to the selection threshold} We modify the threshold $u$ which is 100 mm for Kingston area. Premium decreases with $u$ in a heavy tail model. 

%\paragraph{Sensibilité par rapport à la fonction de vulnérabilité} Nous calibrons le paramètre C de la fonction de vulnérabilité sur les pertes historiques recensées sur les catastrophes majeures (\ref{tab1}). \\
\paragraph{Sensitivity to the vulnerability function} We calibrate the parameter C of vulnerability function on historical losses recorded on major disasters (see table \ref{tab1}). The slope calibration in Figure \ref{fig: vulnerability historical} gives the slope calibration on historical data. \\ 
\begin{figure}[t]
  \centering
   \includegraphics[scale=0.3]{Figures/Chapter3/vul_histo.pdf}
     \rule{35em}{0.5pt}
  \caption{Slope calibration on historical data}
  \label{fig: vulnerability historical}
\end{figure}
%Nous calculons à nouveau la prime pure. Cette dernière est insensible au paramètre C. Dans la même idée, il est possible de modifier le paramètre B. La prime varie de 0,04\% lorsque B varie de 800 mm à 10 000 mm. 

Upon recalculation, the premium appears insensitive. Similarly, changing the parameter shows that the premium increases by 0.04 \% when B varies from 800 mm to 10,000 mm.

%\section*{Transfert de risque}
\section*{Risk transfer}

%\paragraph{Augmentation de capitaux propres}Le but du CCRIF est de rendre accessible son produit contre les pluies torrentielles. Ce dernier peut réduire les coûts commerciaux de la prime en faisant appel à des bailleurs de fonds. Ce mécanisme appliqué la couverture des Ouragans a permis au CCRIF de diviser la prime réclamée au pays souscripteur par deux. 
%Le but du CCRIF est de rendre accessible son produit contre les pluies torrentielles. Ce dernier peut réduire les coûts commerciaux de la prime en faisant appel à des bailleurs de fonds. Ce mécanisme appliqué à la couverture contre les Ouragans a permis au CCRIF de diviser la prime réclamée au pays par deux. \\ \\

\paragraph{Increase in equity}
The purpose of CCRIF is to make affordable XSR product to Carribean countries. It can reduce premium costs by matching donor contributions to XSR underwriting pool. This mechanism applied to Hurricanes coverage policy allows the CCRIF to decrease the premium by two times.

%\paragraph{Réassurance} Afin de limiter les fluctuations du bilan financier liées à la sur-sinistralité des catastrophes naturelles, le CCRIF peut céder des risques à un réassureur comme Swiss Re. L'information sur l'indice paramétrique est partagée entre l'assureur et le réassureur ce qui élimine le risque de base. Des tranches de risques sont tarifiées sur base du modèle de l'indice paramétrique. 
\paragraph{Reinsurance} To reduce the fluctuations of balance sheet related to payouts on major disasters, the CCRIF may transfer risk to a reinsurer as Swiss Re. Information on the parametric index is shared between the insurer and the reinsurer, therefore removing the basis risk. The price of each slice of risk is calculated out by the model on parametric index. 
%\paragraph{Titrisation et Cat-Bonds} 
\paragraph{Securitization and Cat Bonds}
%Pour le CCRIF, la titrisation apparaît comme un excellent moyen de transférer les risques sur les marchés financiers qui ont une capacité d'absorption de pertes plus importante que le marché de l'assurance en cas de sinistre. 
For the CCRIF, securitization appears to be an excellent means of transferring risk to the financial markets that have a capacity to absorb larger losses in major disaster than the insurance market. 
%Dans une structure comme le CCRIF, la titrisation peut apparaitre comme un excellent moyen pour transférer les risques sur les marchés financiers qui ont une capacité d’absorption de pertes plus importante en cas de sinistre
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